#9720371

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#50323	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,9)(3,12,14,8)(4,10,15,5)(6,13,16,11)(17,18,22,20)(19,23,27,24)(21,28,29,25)(26,31,32,30)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3,7,14)(2,8,9,12)(4,13,15,11)(5,16,10,6)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,9)(4,15)(8,12)(11,13)(17,19,22,27)(18,24,20,23)(21,26,29,32)(25,31,28,30)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,4)(2,10)(3,13)(5,9)(6,8)(7,15)(11,14)(12,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,5,7,10)(2,4,9,15)(3,6,14,16)(8,11,12,13)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,6)(2,11)(3,5)(4,8)(7,16)(9,13)(10,14)(12,15)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9627567);